L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 12-s + 2·13-s + 2·14-s + 15-s + 16-s − 2·17-s − 18-s − 20-s + 2·21-s − 2·23-s + 24-s + 25-s − 2·26-s − 27-s − 2·28-s − 4·29-s − 30-s − 4·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.436·21-s − 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.377·28-s − 0.742·29-s − 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.73677631366894, −16.29653034563237, −15.81565565161072, −15.40094804987417, −14.68100796908079, −13.95930877520242, −13.16070681141266, −12.72509977175603, −12.13184068882052, −11.46105654605809, −10.96134734564901, −10.50079238355083, −9.755882766845901, −9.169497014708284, −8.690346017463898, −7.778701133826459, −7.331328884286821, −6.631490060196664, −5.999705097826826, −5.491133673900992, −4.354983871540298, −3.812157200523526, −2.917960971414427, −1.996991471249992, −0.9100079204824594, 0,
0.9100079204824594, 1.996991471249992, 2.917960971414427, 3.812157200523526, 4.354983871540298, 5.491133673900992, 5.999705097826826, 6.631490060196664, 7.331328884286821, 7.778701133826459, 8.690346017463898, 9.169497014708284, 9.755882766845901, 10.50079238355083, 10.96134734564901, 11.46105654605809, 12.13184068882052, 12.72509977175603, 13.16070681141266, 13.95930877520242, 14.68100796908079, 15.40094804987417, 15.81565565161072, 16.29653034563237, 16.73677631366894